On the existence of unstable bumps in neural networks. (English) Zbl 1277.47082
The nonlinear integro-differential equation of Hammerstein is interpreted as modeling the behavior of a single layer of neurons in a neural network, the solution representing the local activity of a neuron and the integral kernel the coupling between neurons. Under some special assumptions on the integral kernel as well as on the nonlinear function of the neurons’ firing rate, the existence of unstable bump solutions to the stated model is proved. Actually, common kernels as ‘Mexican hat’ function or Gauss function do satisfy the conditions. Under additional assumptions, the referred bump solution is a Lyapunov-unstable equilibrium of the model, belonging to the space of continuous functions vanishing at infinity.
Reviewer: Claudia Simionescu-Badea (Wien)
MSC:
| 47N60 | Applications of operator theory in chemistry and life sciences |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 82C32 | Neural nets applied to problems in time-dependent statistical mechanics |
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 47H10 | Fixed-point theorems |
| 37C75 | Stability theory for smooth dynamical systems |
| 45J05 | Integro-ordinary differential equations |
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